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Adaptive and robust radiation therapy optimization for lung cancer

Timothy C. Y. Chan & Velibor V. Mišić

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Chan TC, Mišić VV. Adaptive and robust radiation therapy optimization for lung cancer. European Journal of Operational Research. 2013 Dec 16;231(3):745-56.

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Adaptive and robust radiation therapy optimization forlung cancer

Timothy C. Y. Chana, Velibor V. Misicb,∗

aDepartment of Mechanical and Industrial Engineering, University of Toronto; 5 King’sCollege Road, Toronto, ON M5S 3G8, Canada

bOperations Research Center, Massachusetts Institute of Technology; 77 MassachusettsAvenue, Building E40, Cambridge, MA 02139, United States of America

Abstract

A previous approach to robust intensity-modulated radiation therapy (IMRT)treatment planning for moving tumors in the lung involves solving a singleplanning problem before the start of treatment and using the resulting solu-tion in all of the subsequent treatment sessions. In this paper, we develop anadaptive robust optimization approach to IMRT treatment planning for lungcancer, where information gathered in prior treatment sessions is used to up-date the uncertainty set and guide the reoptimization of the treatment for thenext session. Such an approach allows for the estimate of the uncertain effect toimprove as the treatment goes on and represents a generalization of existing ro-bust optimization and adaptive radiation therapy methodologies. Our methodis computationally tractable, as it involves solving a sequence of linear opti-mization problems. We present computational results for a lung cancer patientcase and show that using our adaptive robust method, it is possible to attainan improvement over the traditional robust approach in both tumor coverageand organ sparing simultaneously. We also prove that under certain conditionsour adaptive robust method is asymptotically optimal, which provides insightinto the performance observed in our computational study. The essence of ourmethod – solving a sequence of single-stage robust optimization problems, withthe uncertainty set updated each time – can potentially be applied to otherproblems that involve multi-stage decisions to be made under uncertainty.

Keywords: OR in health services, intensity-modulated radiation therapy,adaptive radiation therapy, robust optimization, linear programming

∗Corresponding authorEmail addresses: tcychan@mie.utoronto.ca (Timothy C. Y. Chan), vvmisic@mit.edu

(Velibor V. Misic)

Preprint submitted to European Journal of Operational Research May 21, 2013

1. Introduction

Lung cancer is the leading cause of death due to cancer in North America,killing an estimated 180,000 people in 2010 (American Cancer Society, 2010;Canadian Cancer Society’s Steering Committee, 2010) and accounting for over25% of all cancer deaths. Lung cancer is often treated using radiation therapy(Toschi et al., 2007). Of the different types of radiation therapy, one of the mostcommonly used in practice for treating cancer in general is intensity-modulatedradiation therapy (IMRT) (Mell et al., 2005). In an IMRT treatment, the patientis irradiated from multiple beams, each of which is decomposed into a largenumber of small beamlets. The beamlet intensities can be controlled throughthe use of a multileaf collimator (MLC) that moves metal leaves in and out of thebeam field in order to block certain parts of the beam. By appropriately settingthe beamlet intensities, the volume that is irradiated can be made to closelyconform to the shape of the target. The basic problem in planning an IMRTtreatment is to determine how the beamlet intensities or weights should be setso that the target receives an adequate dose while the healthy tissue receives aminimal dose. This is known as the beamlet weight optimization problem or thefluence map optimization problem. Since the inception of IMRT, much researchhas focused on modelling and solving this problem as a mathematical program(see Romeijn & Dempsey, 2008 for a comprehensive overview).

In practice, the beamlet weight optimization problem is complicated by thepresence of uncertainties, such as those arising from errors in beam positioningand patient placement, internal organ motion during treatment, and changesin organ position between treatment sessions. All of these factors affect therelative position of the tumor with respect to the beams, which in turn affectshow much dose is deposited in the tumor and the healthy tissue. For tumors inthe lung, the most significant uncertainty arises from breathing motion. Duringtreatment, the patient is constantly breathing, and the tumor moves with theexpansion and contraction of the patient’s lungs. Furthermore, the patient’sbreathing pattern during treatment is not known exactly beforehand and canvary from day to day. If a treatment is planned with a specific breathing patternin mind but a different pattern is realized during treatment, the tumor may endup being underdosed and the quality of the treatment may thus be greatlycompromised (Lujan et al., 2003; Sheng et al., 2006). At the same time, if thetreatment is designed to deliver the prescription dose to the tumor under anybreathing pattern, regardless of how unlikely some of those patterns may be,then an unnecessarily high amount of damage will be done to the healthy tissue.There exist approaches that aim to mitigate the effect of uncertainty by onlyactivating the beam when the patient is in a certain phase of the breathing cycle(Ohara et al., 1989), but these approaches have the disadvantage of prolongingthe treatment time, leading to reduced throughput in the treatment center.

One methodology that allows for tumor coverage under uncertainty to bebalanced with healthy tissue sparing, without extending the treatment time, isrobust optimization. There are many robust optimization approaches to IMRTtreatment planning. In this paper, we build on the method developed in Bortfeld

2

et al. (2008) and Chan et al. (2006), which is specifically designed to managethe instantaneous breathing motion uncertainty that may be realized duringtreatment sessions. Although robust optimization has been shown to be a valu-able methodology for managing uncertainty due to breathing motion, it can befurther improved by leveraging the fact that treatments are typically fraction-ated. Fractionation refers to the practice of dividing up the prescription doseand delivering small amounts in daily treatment sessions over multiple weeks,in order to exploit the more capable repair mechanism that healthy tissue hasover cancerous tissue (Thames & Hendry, 1987).

To date, the typical approach to fractionated treatment planning and deliv-ery has been to determine what the beamlet intensities should be to deliver acertain prescription dose, scale them by 1/n where n is the number of fractions,and use the resulting intensities in every treatment fraction. This, however,does not take advantage of the dynamics that fractionation injects into IMRTtreatment planning and the underlying uncertainty. For example, it may bethe case that the pre-treatment estimates of the patient’s breathing pattern areerroneous, and the breathing patterns that are planned for differ from thoserealized over the treatment. In this case, the wrong beamlet intensities wouldbe used for every fraction, without the possibility of correcting them. On theother hand, the beamlet intensities may be optimized in such a way that theprescription dose is delivered under a very large range of breathing patterns,when in actuality the patient’s breathing pattern does not vary significantlyfrom fraction to fraction. In this case, there would be no way to correct theconservatism of the treatment, and the patient’s healthy tissue will incur moredamage than necessary. Another example comes from the possibility that thepatient’s breathing pattern may change over the course of treatment as the pa-tient becomes more comfortable in the treatment room or as the evolution ofthe underlying disease affects the patient’s respiratory ability. As a result, lat-ter treatment sessions may deliver insufficient dose to the tumor and more dosethan anticipated to healthy tissue. These issues motivate the incorporation ofadaptive re-optimization into robust optimization models.

To deal with fractionation in more general IMRT treatment planning con-texts, much research has been conducted in the medical physics community inwhat is known as off-line adaptive radiation therapy (ART). In off-line ART,after the current fraction is delivered, new information, typically obtained fromimaging devices, is fed back into the planning process to design the beamletintensities of the next fraction (Yan et al., 1997a). To date, off-line ART androbust optimization have largely been considered independently of each other.

In this paper, we propose a novel method that generalizes the frameworkof off-line adaptive radiation therapy and the robust optimization methodologyof Bortfeld et al. (2008) and Chan et al. (2006) to treat lung cancer in thepresence of breathing motion uncertainty. Our method is based on adaptivelyupdating an uncertainty set that models the breathing motion uncertainty andthen solving a robust optimization problem with the new uncertainty set prior toeach fraction. As a result, our method is able to combat both the instantaneousuncertainty realized in each fraction as well as changes in this uncertainty that

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occur from day to day. The clinical value of this method is that it has thepotential to improve upon the non-adaptive robust optimization method in bothtumor coverage and healthy tissue sparing. This is significant because patientsurvival rates have been shown to improve when the tumor dose is increased (seePerez et al., 1986). Unfortunately, dose escalation to the tumor is limited by theradiation tolerance of healthy tissue, particularly the lungs (Cox et al., 1990).If it is possible to increase tumor dose and reduce healthy tissue dose from thelevels achieved by non-adaptive robust optimization, further dose escalation willbecome viable, and improvements in patient survival rate may be realized.

Our specific contributions in this paper are as follows:1. We develop a method that combines robust optimization with adaptive

radiation therapy and suitably generalizes both frameworks. This method hasmodest computational requirements and can be easily generalized to other typesof cancer and other types of uncertainty.

2. Using real patient data, we demonstrate computationally that this methodgenerates treatments that improve upon treatments obtained by non-adaptiverobust methods in both tumor coverage and lung dose. We show that thequality of the final treatment does not vary significantly with the choice of initialuncertainty set. We also show that the achieved performance is comparableto that of two “prescient” solutions, which correctly anticipate the patient’sbreathing over the entire treatment.

3. We prove an asymptotic optimality result: if the patient breathing patternconverges over the treatment course, then the dose distribution produced by ouradaptive robust method converges to an “ideal” set of dose distributions thatexhibit no tumor underdose or overdose and low healthy tissue dose. We usethis result to explain multiple phenomena observed in our computational study.

The rest of the paper is organized as follows. In Section 2, we briefly reviewsome of the relevant literature in robust optimization and off-line ART. In Sec-tion 3, we provide an overview of the non-adaptive robust optimization methodon which this paper builds. In Section 4, we provide a detailed description ofthe proposed adaptive robust method. In Section 5, we provide results fromour computational study with clinical lung cancer patient data. In Section 6,we conduct a theoretical analysis of the adaptive robust method. Finally, inSection 7, we provide some conclusions and directions for future work.

2. Literature Review

Robust optimization has been extensively developed in the OR community(see Ben-Tal et al., 2009 and Bertsimas et al., 2011), and has received increas-ing focus in the last decade as a methodology for IMRT treatment planningunder uncertainty. In Chu et al. (2005), tumor position uncertainty due tointerfraction motion (organ position changes that occur between treatment ses-sions) and setup error was modelled using an ellipsoidal uncertainty set andthe authors showed that the resulting robust solution achieves the same level oftumor coverage as a clinical margin solution but with less healthy tissue dose.

4

In Olafsson & Wright (2006), the uncertainty due to dose calculation error andinterfraction organ motion was also modelled using an ellipsoidal uncertaintyset, and the authors showed for a nasopharyngeal case that a robust solutionachieves better tumor coverage than the nominal solution (one which assumesa dose matrix known with certainty) and leads to better organ sparing than theclinically prescribed margin solution. In Chan et al. (2006) and Bortfeld et al.(2008), uncertainty due to intrafraction breathing motion was modelled using apolyhedral uncertainty set, and it was shown that the robust solution providesimproved tumor coverage over the nominal solution, while providing tumor cov-erage comparable to the margin solution at reduced healthy tissue dose. All ofthe robust IMRT optimization methods described above are non-adaptive.

With regard to off-line ART, a variety of procedures have been proposed forhow to treat a patient based on information obtained in the most recent fraction.Many studies have proposed using daily computed tomography (CT) images asa form of feedback. For example, Mohan et al. (2005), Lu et al. (2006), Wuet al. (2006) and Wu et al. (2008) all studied schemes where daily CT imagesare used to periodically assess the treatment plan for errors in dose deliverydue to organ deformation and intrafraction motion (organ motion that occursduring a treatment session), and to aid either the re-adjustment or complete re-optimization of the treatment for subsequent fractions. Other imaging methodshave also been proposed for off-line ART. For instance, de la Zerda et al. (2007)proposed algorithms for generating new treatment plans from fraction to fractionin response to changes in the patient geometry detected from cone beam CT(CBCT) images, as well as to the cumulative delivered dose. Another exampleis Yan et al. (1998), who tested an ART method where the daily setup error ismeasured using portal imaging and is used to correct the treatment if the errorbecomes sufficiently large.

Some ART studies have also incorporated probabilistic models of uncer-tainty into their proposed planning methods. Yan et al. (1997a) studied anART method where the daily treatment target and beam placement variationare measured and used to modify the treatment dose and field margin in eachfraction under the assumption that the error is normally distributed. Lof et al.(1998) studied an ART scheme where the daily setup position is measured in ev-ery fraction, but assumed that the probabilistic dynamics of the internal organmotion are known beforehand. Rehbinder et al. (2004) applied linear quadraticcontrol to ART planning and showed that using this form of adaptation removesthe need for a margin, allowing for safer dose escalation. The problem of ARTcan also be seen as a problem of sequential decision making under uncertaintyand as such, several studies have considered dynamic programming techniques.For example, Sir et al. (2010) showed that an adaptive open-loop feedbackcontrol (OLFC) policy achieves better performance than a non-adaptive OLFCpolicy. Ferris & Voelker (2004) and Deng & Ferris (2008) applied neuro-dynamicprogramming (NDP) to ART and similarly showed that their NDP policy offersan improvement over a constant, non-adaptive policy.

The limitation of the majority of algorithms for off-line ART is in how theyhave approached uncertainty. Many studies have not explicitly included uncer-

5

tain effects into their treatment planning process. Of the studies that have, mosthave assumed that realizations of the uncertain effect from fraction to fractionare independent and identically distributed – although this may be appropriatefor setup error and patient positioning, it may not be suitable for other typesof uncertainty. The most significant assumption present in studies that haveincorporated probabilistic models of uncertainty is that the underlying proba-bility distribution of the uncertain effect is known precisely before the start ofthe treatment. In actuality, the distribution which underlies an uncertain effectfor a particular patient in a particular treatment is never known precisely, andas already discussed, the quality of a treatment designed with a specific proba-bility distribution in mind can deteriorate significantly if a different distributionis realized.

The fundamental difference between the approach proposed in this paperand the majority of prior ART work lies in the assumptions that are madeabout the uncertainty. In our approach, we do not assume that the nature ofthe uncertainty is known precisely a priori nor that it needs to stay constantfrom fraction to fraction. There has been some prior research into using previ-ous measurements of uncertainty to update an estimate of the uncertain effect,using such methods as Kalman filtering (Yan et al., 1997b; Keller et al., 2003)and Bayesian updating (Lam et al., 2005; Sir, 2007). Very few studies have con-sidered these methods with treatment re-planning (Yan et al., 1997b and Sir,2007 are exceptions) and these studies assume that the uncertain effect followsa particular distribution. No studies have combined such estimation methodswith the kind of robust optimization approach we consider.

3. Static robust IMRT optimization

In this section, we review the robust optimization approach of Chan et al.(2006) and Bortfeld et al. (2008). In their approach, the planner decides onan uncertainty set before treatment and solves the robust optimization problemcorresponding to that uncertainty set. The optimal solution of the robust prob-lem is a vector that specifies the intensity of each beamlet in the ensemble ofbeams to be used for treatment. These optimized beamlet intensities are robustto the uncertain effect whenever it takes on values from the uncertainty set:whenever the uncertain effect realizes a value inside the uncertainty set whilethe patient is being irradiated according to the optimized intensities, the doseto every part of the tumor will be within the prescribed bounds. We will referto this approach as the static robust optimization approach, as the beamlet in-tensity vector is fixed and does not change over the course of the treatment (ineach fraction, 1/nth of the intensity vector is delivered, where n is the numberof fractions). The patient’s breathing motion is modelled using a probabilitymass function (PMF). The breathing motion PMF for a patient in a particularfraction specifies the proportion of time the patient spends in each of a finitenumber of breathing motion states during that fraction. Each breathing mo-tion state corresponds to a particular displacement or snapshot of the patientgeometry during the patient’s respiratory cycle. The uncertainty set is then

6

a set of breathing motion PMFs that could be realized during treatment. Wenote here that in the method of Bortfeld et al. (2008), the set of breathing mo-tion states is determined from pre-treatment breathing-correlated 4D computedtomography (4D-CT) images. This assumes that the amplitude of the tumormotion observed in these pre-treatment images represents the full extent of thetumor motion, and that the amplitude of the motion that will be observed dur-ing the ensuing treatment sessions will not exceed these limits. This potentialissue can be circumvented by having the patient take deep breaths during thepre-treatment 4D-CT imaging session, allowing the treatment planner to moreaccurately construct the set of breathing motion states.

We now define the static robust IMRT optimization problem. Let B bethe set of beamlets and let wb denote the intensity of beamlet b ∈ B; the wb

variables are the decision variables of the robust optimization problem. Thepatient geometry (the healthy organs and tumor tissue) is divided into threedimensional voxels, or volume elements. Let V denote the set of all voxels andT denote the set of tumor voxels. Let X denote the finite set of breathingmotion states. Each breathing motion state is associated with a particulardisplacement of the patient geometry (the voxels), so to each beamlet b ∈ B,breathing motion state x ∈ X and voxel v ∈ V, we associate a dose depositioncoefficient ∆v,x,b. This coefficient specifies the amount of dose deposited in voxelv when the patient is in breathing motion state x and the beamlet b is at unitintensity. Let p(x) be the probability of the patient being in motion state x atany given time during a treatment session. The dose that is delivered to voxelv is then given by ∑

x∈X

∑b∈B

∆v,x,bp(x)wb,

which is the sum of the doses to voxel v that would be delivered under eachbreathing motion state weighted by the corresponding proportions of time spentin those states. Due to the use of beamlet intensities, this model does notaccount for the interplay effect between the motion of the MLC leaves and thetumor motion (see, for example, Yu et al., 1998 and Bortfeld et al., 2002); theimpact of such an effect, however, is small in practice (Bortfeld et al., 2004).For each tumor voxel v ∈ T , let θv be the prescribed minimum dose and let γθvbe the prescribed maximum dose, where γ ≥ 1.

Let P be the set of all PMFs on the finite set X, defined as

P =

{p ∈ R|X| For each x ∈ X, p(x) ≥ 0;

∑x∈X

p(x) = 1

}.

The set P defines the (|X| − 1)-dimensional unit simplex. Let P ⊆ P be theuncertainty set, which is a bounded polyhedron defined by a lower bound vector` and an upper bound vector u:

P = {p ∈ P For each x ∈ X, `(x) ≤ p(x) ≤ u(x)} .

7

With these definitions, the static robust problem from Bortfeld et al. (2008)is

minimize∑v∈V

∑x∈X

∑b∈B

∆v,x,bp(x)wb

subject to∑x∈X

∑b∈B

∆v,x,bp(x)wb ≥ θv, ∀v ∈ T , p ∈ P,∑x∈X

∑b∈B

∆v,x,bp(x)wb ≤ γθv, ∀v ∈ T , p ∈ P,

wb ≥ 0, ∀b ∈ B,

(1)

where p is some nominal PMF chosen by the treatment planner before the startof treatment as approximately representative of the patient’s overall breathingpattern. The optimal solution of (1) is a beamlet intensity vector w∗ that meetsminimum and maximum tumor dose requirements no matter which breathingmotion PMF p ∈ P is realized, while ensuring the lowest possible total doseto the patient under the nominal PMF. In the case that p /∈ P , it was shownempirically in Bortfeld et al. (2008) that w∗ remains feasible with high prob-ability. Due to the linearity of this model and the use of polyhedral uncer-tainty sets, this model is capable of accomodating robust versions of many othertypes of clinically relevant constraints, such as partial volume constraints basedon conditional-value-at-risk (see Romeijn et al., 2006) or constraints involvingαEUD (Thieke et al., 2002), which is a linear approximation to the popularequivalent uniform dose (EUD) metric. Our goal in this paper is to elucidatethe potential benefits of a framework that combines adaptation and robustness,not to design clinic-ready solutions. We therefore consider the stylized modelabove, which only enforces lower and upper dose bounds on the tumor.

There are two special cases of (1) that will be referred to later. The firstis the case when P consists of a single PMF. The resulting problem is calledthe nominal problem; when P = {p}, we refer to problem (1) as the nominalproblem with respect to p. The solution of the nominal problem is the leastconservative treatment we can deliver, as we are assuming that the PMF thatwill be realized during treatment will be the exact PMF that we are guardingagainst. The second is the case when P consists of all possible PMFs on X,that is, P = P; this problem is called the margin problem. The solution ofthe margin problem is the most conservative treatment that we can deliver,because we assume that any PMF is possible. These two types of formulationsrepresent two extremes of a continuum of robustness (Chan et al., 2006), withthe nominal problem being the least robust and the margin problem being themost robust. The robust problem with a general uncertainty set (which isneither a singleton nor P) is somewhere in between. With increasing levels ofrobustness, characterized by larger uncertainty sets, the level of tumor coverageincreases, but at the cost of increased dose to healthy tissue.

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4. Adaptive robust IMRT optimization

In the adaptive robust approach that we propose, we allow the beamlet in-tensities to be re-optimized from fraction to fraction in the following way. Ona given day, the patient is irradiated with a beamlet intensity vector obtainedby solving (1) with the current uncertainty set as input. During or immediatelyafter the fraction is delivered, the patient’s breathing motion PMF is measured.This measurement, together with the current uncertainty set, is used to gen-erate a new uncertainty set. The robust problem is then solved with this newuncertainty set, leading to a new beamlet intensity vector to be used on the nextday. This process is repeated for each fraction until the end of the treatment.This procedure is presented as Algorithm 1.

Although in theory one could measure the patient’s PMF before the deliveryof a fraction and plan the fraction using the resulting PMF, this approach wouldconstrain the amount of time that would be available for treatment planning andquality assurance, as the patient would have to wait between measurement andfraction delivery. By measuring the patient’s PMF during or after the deliveryof the fraction, this temporal constraint can be avoided by having treatmentplanning and quality assurance done offline between treatment fractions, whichwould be more desirable from an operational point of view. Furthermore, wewill show in Section 6 that these two methods have the same asymptotic per-formance.

Algorithm 1 Adaptive robust optimization method

Require: Total number of fractions n, initial uncertainty set P 1

1: Initialize i = 12: Solve problem (1) with P = P 1 to obtain a beamlet intensity vector w1

3: Deliver w1/n to the patient4: while i ≤ n− 1 do5: Measure the patient’s breathing motion and construct the PMF pi

6: Generate the new uncertainty set P i+1 from P i, pi

7: Solve problem (1) with P = P i+1 to obtain a new beamlet intensity vectorwi+1

8: Deliver wi+1/n to the patient9: Set i = i+ 1

10: end while

4.1. Update algorithms

One major step in the adaptive robust approach specified as Algorithm 1is the generation of the new uncertainty set from the old uncertainty set andthe most recent measurement of the patient’s breathing motion PMF. We referto any procedure that can be used to perform this step as an uncertainty setupdate algorithm. In our computational study we consider two such updatealgorithms, exponential smoothing and running average, though our theoreticalanalysis generalizes to a larger class of update algorithms.

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Recall that we define an uncertainty set P in terms of a lower bound vector` and an upper bound vector u. Therefore, the uncertainty set for fraction i,denoted P i, is defined by lower and upper bound vectors `i and ui, respectively.The update algorithms we present will determine lower and upper bound vectors`i+1,ui+1 from the most recent vectors `i,ui and the most recent PMF pi.

The exponential smoothing update algorithm generates a new uncertaintyset by taking a convex combination of the most recent lower and upper boundvectors with the most recent PMF. Specifically, given `i, ui and pi for day i,`i+1 and ui+1 are defined as

`i+1(x) = (1− α)`i(x) + αpi(x), (2)

ui+1(x) = (1− α)ui(x) + αpi(x), (3)

for each x ∈ X, where α ∈ [0, 1].The running average update algorithm generates a new uncertainty set by

averaging the PMFs that have been observed so far, together with the very firstlower and upper bound vectors. Specifically, given `1, u1 and p1, . . . ,pi on dayi, `i+1 and ui+1 are defined as

`i+1(x) =1

i+ 1

`1(x) +

i∑j=1

pj(x)

,

ui+1(x) =1

i+ 1

u1(x) +

i∑j=1

pj(x)

.

for each x ∈ X. These expressions can be rewritten as functions of only `i(x),ui(x) and pi(x):

`i+1(x) =i

i+ 1`i(x) +

1

i+ 1pi(x), (4)

ui+1(x) =i

i+ 1ui(x) +

1

i+ 1pi(x). (5)

4.2. Prescient solutions

Because the update algorithms are myopic and are not guaranteed to cor-rectly anticipate the patient’s breathing motion PMF in every fraction, it isimportant to be able to measure the loss of optimality with respect to a “pre-scient” solution – one that would be generated if we knew p1, . . . ,pn ahead oftime.

We consider two kinds of prescient solutions. The first type of prescientsolution that we define is the daily prescient solution. This solution is obtainedby setting each wi to the solution of problem (1) with P = P i = {pi}. Itis clear that using this solution, the dose to each tumor voxel v is betweenθv/n and γθv/n in each fraction, which ensures that the cumulative dose toevery tumor voxel v is within the prescribed bounds θv and γθv. Furthermore,

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because each wi is obtained from a nominal problem, the overall dose deliveredto the patient should be low compared to other treatments that also deliver theminimum required dose to every tumor voxel.

The second type of prescient solution that we define is the average prescientsolution. This solution is obtained by solving the nominal problem with P ={pavg}, where pavg = 1/n ·

∑ni=1 p

i and setting each wi to the correspondingoptimal solution w of this problem. It can be easily verified that under thissolution, the final dose to every tumor voxel is within the prescribed boundsfor that voxel. Furthermore, because the intensity vector of every fraction isobtained from a nominal problem, the overall dose delivered to the patientshould be low compared to other treatments that also deliver the minimumrequired dose to every tumor voxel. While each tumor voxel v ∈ T is notguaranteed to receive θv/n in each fraction, the fact that wi = wj for any twofractions i and j ensures that the dose delivered by this solution in any fractioncannot be too low.

5. Computational study

5.1. Background

In order to compare our results with previous static robust optimization re-sults, the patient geometry and beam geometry used in this paper were exactlythe same as in Bortfeld et al. (2008). The patient geometry consisted of 110,275voxels, of which 5,495 were tumor voxels, and each voxel was of size 2.93 mm ×2.5 mm × 2.93 mm. The set of tumor voxels for this patient case correspondedto the clinical target volume (CTV). The robust problem solved in each fractionof the adaptive method had 122,515 variables and 65,940 constraints. The pa-rameter γ was set to 1.1, while the minimum dose θv was set to 72Gy for everytumor voxel v. The set X consisted of five motion states. The nominal PMFused in the objective function was the same nominal PMF used previously. Theadaptive robust and the static robust methods were tested using two differentPMF sequences derived from clinical lung cancer patient data obtained at Mas-sachusetts General Hospital; details on these PMF sequences can be found inSection A of the Online Supplement.

The two PMF sequences consisted of the first 30 PMFs from the two ex-periments, respectively, used in Bortfeld et al. (2008). This number of PMFswas chosen to simulate a realistic six-week, Monday-to-Friday treatment course.In the discussion that follows, we will focus on the first PMF sequence, as theresults obtained from the second PMF sequence were qualitatively very similar;for completeness, results for the second PMF sequence are provided in Section Bof the Online Supplement. For the adaptive robust method, we tested the ex-ponential smoothing update algorithm with α equal to 0.1, 0.5, 0.9 and 1, andthe running average update algorithm. The specific values of α for the expo-nential smoothing method were chosen to demonstrate the performance of theapproach at varying speeds of adaptation (as α increases, the uncertainty setis adjusted more rapidly to follow the realized PMFs). Three different initial

11

uncertainty sets were used for the adaptive and static robust methods. The firsttype of uncertainty set was a nominal uncertainty set, with ` and u both equalto the nominal PMF. The second uncertainty set was the margin uncertaintyset (P = P, the entire unit simplex) – for this patient case, the margin uncer-tainty set treatment corresponds to treating the internal target volume (ITV),which is the volume obtained by taking the union of the tumor volume at allof the different phases of the breathing cycle. The third uncertainty set was arobust (intermediately-sized) uncertainty set, which was neither a singleton northe whole space of PMFs P; for each PMF sequence we used the correspondingrobust uncertainty set from Bortfeld et al. (2008). In addition to the adaptiveand static robust methods, both prescient solutions were calculated for each ofthe two PMF sequences.

The experiments were performed on a Dell workstation with a quad-core IntelXeon 2.67GHz processor and 6GB of memory. All of the steps of the adaptiverobust method – with the exception of the step where the robust problem issolved to obtain wi+1 – were implemented in MATLAB (The MathWorks, Inc.,Natick, Massachusetts). The robust problem in each fraction was solved usingCPLEX 12.1 (IBM Corp., Armonk, NY) via AMPL (AMPL Optimization LLC,Albuquerque, NM) and took no more than 20 minutes to solve in the majorityof cases using CPLEX 12.1’s barrier method.

After the adaptive and static robust methods were executed with each ofthe PMF sequences described above, the resulting treatments were evaluated bycalculating the final dose distribution. The final dose distribution is a vector d ∈R|V| of doses delivered to the voxels in the patient geometry, defined component-wise as

dv =

n∑i=1

∑x∈X

∑b∈B

∆v,x,bpi(x)

wib

n.

From the final dose distribution, we calculated three key statistics: the minimumtumor dose, dmin

T = minv∈T dv; the mean left lung dose, dmeanL =

∑v∈L dv/|L|,

where L is the set of voxels belonging to the left lung; and the mean normaltissue dose, dmean

N =∑

v∈N dv/|N |, where N is the set of normal tissue (non-tumor) voxels. In addition to these statistics, we also compute two other metrics.The mean left lung dose-scaled (MLLD-scaled) tumor dose of a treatment is theminimum dose to the tumor when the dose to the patient is scaled so thatthe mean left lung dose of the treatment matches the mean left lung dose ofthe static margin treatment. Similarly, the left lung V20-scaled (LLV20-scaled)tumor dose of a treatment is the minimum dose to the tumor when the dose tothe patient is scaled so that the lung V20 value (the volume of the lung receiving20Gy or more) of the treatment matches the lung V20 value of the static margintreatment.

5.2. Results

Table 1 displays dose statistics for the first PMF sequence. In Figures 1 and2, we plot the minimum tumor dose as a percentage of 72Gy versus the mean leftlung dose as a percentage of the static margin solution for each method applied

12

85 90 95 100

93

94

95

96

97

98

99

100

101

Mean left lung dose as percentage of static margin dose

Min

imum

tum

or d

ose

as p

erce

ntag

e of

72G

y

Minimum tumor dose versus mean left lung dose

(S,N)

(S,R)

(S,M)

StaticExponential smoothing, α = 0.1Exponential smoothing, α = 0.5Exponential smoothing, α = 0.9Exponential smoothing, α = 1Running averageDaily prescientAverage prescient

Figure 1: Plot of minimum tumor dose versus mean left lung dose for the different implemen-tations applied to the first PMF sequence (full). The points in the bottom left part of the plotare the adaptive robust and prescient treatments. Point labels follow the notation of Table 1.

to the first PMF sequence. Figure 1 shows all of the treatments, while Figure 2zooms in on the adaptive and prescient treatments. On these plots, pointsobtained by the same method (static, exponential smoothing with parameterα or running average) from different initial uncertainty sets are joined togetherby straight lines to illustrate the approximate Pareto frontiers generated by thecorresponding method.

In Figure 3, we also provide dose volume histograms (DVHs) for two treat-ments obtained from the first patient sequence – the exponential smoothing al-gorithm with smoothing factor 0.5 and robust uncertainty set (implementation(ES(0.5),R)) and the static method with the same uncertainty set (implemen-tation (S,R)). A DVH is a plot for a particular structure which indicates, for agiven level of dose, what volume of the structure receives at least that dose. Ide-ally, the healthy organ DVHs would be step functions from 100% to 0% at 0Gy(i.e., no dose is delivered to the healthy organs), and the tumor DVH would bea step function from 100% to 0% at 72Gy (i.e., the whole tumor volume receivesexactly 72Gy).

5.3. Discussion

The computational results generate three important insights. First, theadaptive robust method generally dominates the static robust method, gener-ating solutions that can simultaneously improve on tumor coverage and healthytissue dose. As an example of this, compare the dose statistics of the staticand the exponential smoothing with α = 0.5 methods for the first PMF se-quence in Table 1. For the margin (M) uncertainty set, moving from the static(implementation (S,M)) to the exponential smoothing with α = 0.5 (imple-mentation (ES(0.5),M)) treatment results in a reduction in the mean left lung

13

Tab

le1:

Dose

stati

stic

sfo

rth

efi

rst

PM

Fse

qu

ence

.

Imp

lem

enta

tion

1M

in.

tum

ord

ose

Mea

nlu

ng

dose

Mea

nn

.ti

ssu

ed

ose

ML

LD

-sca

led

tum

or

dose

LLV

20-s

cale

dtu

mor

dose

Gy

%2

Gy

%3

Gy

%4

Gy

%5

Gy

%6

(S,N

)67

.25

93.4

017

.39

85.2

69.0

488.9

978.8

7109.4

882.1

4114.0

2(E

S(0

.1),

N)

70.7

998

.32

17.5

085.8

59.0

589.1

182.4

6114.4

688.4

2122.7

3(E

S(0

.5),

N)

71.7

499

.63

17.5

786.1

69.0

689.1

383.2

6115.5

790.4

5125.5

5(E

S(0

.9),

N)

71.8

399

.77

17.5

786.1

79.0

689.1

383.3

6115.7

190.7

1125.9

2(E

S(1

),N

)71

.84

99.7

717

.57

86.1

69.0

689.1

383.3

8115.7

490.7

8126.0

1(R

A,N

)71

.47

99.2

617

.52

85.9

19.0

689.1

283.1

9115.4

790.1

0125.0

6

(S,R

)71

.37

99.1

318

.19

89.2

09.4

492.9

380.0

1111.0

683.8

8116.4

3(E

S(0

.1),

R)

71.8

099

.72

17.8

387.4

39.1

990.4

682.1

2113.9

887.5

5121.5

3(E

S(0

.5),

R)

71.9

999

.98

17.6

586.5

59.0

989.5

083.1

8115.4

690.2

8125.3

2(E

S(0

.9),

R)

71.9

899

.97

17.6

086.3

39.0

789.2

883.3

8115.7

390.7

3125.9

5(E

S(1

),R

)71

.97

99.9

517

.59

86.2

99.0

789.2

683.4

0115.7

790.8

0126.0

3(R

A,R

)71

.88

99.8

417

.66

86.6

19.1

289.7

182.9

9115.2

089.6

3124.4

1

(S,M

)72

.04

100.

0620

.39

100.0

010.1

6100.0

072.0

4100.0

072.0

4100.0

0(E

S(0

.1),

M)

72.0

510

0.07

18.3

189.7

89.4

492.9

180.2

5111.4

082.4

5114.4

5(E

S(0

.5),

M)

72.0

310

0.04

17.8

087.2

79.1

690.1

082.5

3114.5

688.8

6123.3

5(E

S(0

.9),

M)

72.0

210

0.02

17.7

186.8

79.1

189.6

782.9

0115.0

789.8

7124.7

5(E

S(1

),M

)71

.99

99.9

817

.67

86.6

59.0

989.4

983.0

8115.3

290.3

2125.3

7(R

A,M

)72

.03

100.

0417

.94

87.9

79.2

390.8

681.8

8113.6

686.3

6119.8

8

(DLY

P)

72.0

010

0.00

17.5

786.1

89.0

689.1

383.5

5115.9

791.1

6126.5

4

(AV

GP

)72

.00

100.

0017

.54

86.0

29.0

689.1

383.7

0116.1

991.5

9127.1

3

1U

nd

er“Im

ple

men

tati

on

”,

the

firs

tte

rmin

dic

ate

sth

ety

pe

of

met

hod

:st

ati

c(S

),ex

pon

enti

al

smooth

ing

wit

hp

ara

met

erα

(ES

(α))

,ru

nn

ing

aver

age

(RA

),d

aily

pre

scie

nt

(DLY

P)

an

daver

age

pre

scie

nt

(AV

GP

).F

or

the

non

-pre

scie

nt

met

hod

s,th

ese

con

dte

rmin

dic

ate

sw

hic

hof

the

un

cert

ain

tyse

tsd

escr

ibed

inS

ecti

on

5.1

was

use

das

the

init

ial

un

cert

ain

tyse

t:n

om

inal

(N),

rob

ust

(R)

an

dm

arg

in(M

).2

Th

ep

erce

nta

ge

un

der

“M

in.

tum

or

dose

”is

the

min

imu

mtu

mor

dose

as

ap

erce

nta

ge

of

the

pre

scri

pti

on

dose

(72G

y).

3T

he

per

centa

ge

un

der

“M

ean

lun

gd

ose

”is

the

mea

nle

ftlu

ng

dose

as

ap

erce

nta

ge

of

the

mea

nle

ftlu

ng

dose

del

iver

edin

the

stati

cm

arg

intr

eatm

ent,

ind

icate

das

imp

lem

enta

tion

(S,M

).4

Th

ep

erce

nta

ge

un

der

“M

ean

n.

tiss

ue

dose

”is

defi

ned

an

alo

gou

sly

toth

ep

erce

nta

ge

un

der

“M

ean

lun

gd

ose

”.

5T

he

per

centa

ge

un

der

“M

LL

D-s

cale

dtu

mor

dose

”is

the

min

imu

mtu

mor

dose

as

ap

erce

nta

ge

of

the

stati

cm

arg

inm

inim

um

tum

or

dose

wh

enth

etr

eatm

ent

inqu

esti

on

issc

ale

dto

have

the

sam

em

ean

left

lun

gd

ose

as

the

stati

cm

arg

intr

eatm

ent.

6T

he

per

centa

ge

un

der

“L

LV

20-s

cale

dtu

mor

dose

”is

the

min

imu

mtu

mor

dose

as

ap

erce

nta

ge

of

the

stati

cm

arg

inm

inim

um

tum

or

dose

wh

enth

etr

eatm

ent

inqu

esti

on

issc

ale

dto

have

the

sam

ele

ftlu

ng

V20

as

the

stati

cm

arg

intr

eatm

ent.

14

85 86 87 88 89 90 91 92

97.5

98

98.5

99

99.5

100

100.5

101

Mean left lung dose as percentage of static margin dose

Min

imum

tum

or d

ose

as p

erce

ntag

e of

72G

y

Minimum tumor dose versus mean left lung dose

(S,R)

(ES(0.1),N)

(ES(0.1),R)

(ES(0.1),M)

(ES(0.5),N)

(ES(0.5),R)

(ES(0.5),M)

(ES(0.9),N)

(ES(0.9),R)

(ES(0.9),M)

(ES(1),N)

(ES(1),R) (ES(1),M)

(RA,N)

(RA,R)

(RA,M)

(DLYP)(AVGP)

StaticExponential smoothing, α = 0.1Exponential smoothing, α = 0.5Exponential smoothing, α = 0.9Exponential smoothing, α = 1Running averageDaily prescientAverage prescient

Figure 2: Plot of minimum tumor dose versus mean left lung dose for the different implemen-tations applied to the first PMF sequence (zoomed-in). Point labels follow the notation ofTable 1.

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

80

90

100

Dose (Gy)

Rel

ativ

e vo

lum

e (%

)

Dose volume histogram − first PMF sequence, (ES(0.5), R) vs. (S,R)

Left lung −− (ES(0.5), R)Left lung −− (S,R)Esophagus −− (ES(0.5), R)Esophagus −− (S,R)Spinal cord −− (ES(0.5), R)Spinal cord −− (S,R)Heart −− (ES(0.5), R)Heart −− (S,R)Tumor −− (ES(0.5), R)Tumor −− (S,R)

Figure 3: Dose volume histogram comparing the treatments obtained by exponential smooth-ing with α = 0.5 started from the robust uncertainty set (implementation (ES(0.5),R)) andthe static method with the robust uncertainty set (implementation (S,R)) with the first PMFsequence.

15

dose of 12.73% of the static margin dose, with virtually no change in tumorcoverage. For the robust (R) uncertainty set, moving from the static (imple-mentation (S,R)) to the exponential smoothing with α = 0.5 (implementation(ES(0.5),R)) treatment results in an increase in minimum tumor dose of 0.85%of the prescribed dose, and a decrease in the mean left lung dose of 2.65% ofthe static margin dose – both objectives are improved. For the nominal (N)uncertainty set, moving from the static (implementation (S,N)) to the exponen-tial smoothing with α = 0.5 (implementation (ES(0.5),N)) treatment results ina significant increase in the minimum tumor dose of 6.23% of the prescriptiondose with only a marginal increase in mean left lung dose of 0.91% of the staticmargin dose. Clinically, the marginal increase in lung dose (0.18Gy) would beacceptable given the increase in minimum tumor dose (4.49Gy). The dominanceof the adaptive robust method over the static robust method is also evident inFigures 1 and 2. On these plots, we can see that the frontiers of all of theadaptive methods are generally below and to the left of the static frontier. TheDVHs in Figure 3 further support this insight. From Figure 3, we can see thatthe left lung, esophagus and heart curves for the adaptive treatment are mostlyto the left of the corresponding curves for the static treatment, indicating anoverall reduction of dose in these structures. The maximum spinal cord dose isslightly higher in the adaptive treatment, but is still clinically acceptable. Thehomogeneity of the tumor dose is slightly better in the static treatment, as canbe seen in the sharper decline of the static treatment tumor dose-volume curve.However, the adaptive treatment has less tumor underdose and still satisfiesthe maximum tumor dose constraint. Again, Figure 3 supports the observationthat the adaptive robust method generally leads to simultaneous improvementin tumor coverage and healthy tissue sparing.

Alternatively, we can analyze these simultaneous improvements from an iso-dose perspective, where the dose delivered to the patient under each treatmentis normalized to match the static margin treatment in certain lung dose metricsand the resulting (scaled) tumor doses are then compared. From this per-spective, our results show that adaptation allows for significant dose escalationbeyond the existing static robust approach. For example, if the prescriptiondose for the static robust treatment (implementation (S,R)) for the first PMFsequence is scaled up so that the mean left lung dose matches that of the staticmargin treatment (implementation (S,M)), the resulting minimum tumor dose(the MLLD-scaled tumor dose in Table 1) is 80.01Gy. Doing the same for theexponential smoothing treatment with α = 0.5 and the robust uncertainty set(implementation (ES(0.5),R)) for the first PMF sequence, the MLLD-scaled tu-mor dose is 83.18Gy – an increase of 3.17Gy ('4% of the original prescriptiondose of 72Gy) from the static robust value. Using the estimated relationshipbetween the 5-year local tumor control rate and tumor dose from Kong et al.(2005) for non-small cell lung cancer (NSCLC), this difference of 3.17Gy trans-lates to a 4% increase in local tumor control. In addition to the mean lung dose,dose can also be escalated according to the commonly used lung V20 metric,which is defined as the volume of the lung tissue that receives a dose of 20Gyor higher; the minimum tumor dose that results from this type of escalation is

16

given under the column labeled “LLV20-scaled tumor dose” in Table 1. If theprescription dose is escalated so that the (ES(0.5),R) and (S,R) treatments forthe first PMF sequence have the same left lung V20 as the (S,M) treatment,then the adaptive robust solution leads to an additional minimum tumor doseescalation of 6.40Gy ('9% of the original prescription dose of 72Gy) beyondthat of the static robust approach. Using the relationship from Kong et al.(2005), this difference of 6.40Gy translates approximately to an 8% increase in5-year local tumor control. Given that the 5-year control rates for tumor dosesin the 63-69Gy and 74-84Gy ranges are on the order of 12% and 35% respec-tively (Kong et al., 2005), the tumor dose escalation and the associated gains inlocal control rate using both the mean left lung dose and left lung V20 methodsthat we have described have the potential to be significant.

The second insight that can be drawn is that even though the adaptive ro-bust method is myopic and does not know what PMFs will be realized in futurefractions, its performance for this patient case comes close to the performanceof solutions that correctly anticipate future PMFs. This is evident in the factthat the treatments obtained from the adaptive robust method are very close intumor coverage and healthy tissue dose to the two types of prescient solutions.For example, for the first PMF sequence, when the robust (R) uncertainty setis chosen and exponential smoothing with α = 0.9 is applied (implementation(ES(0.9),R) in Table 1), the minimum tumor dose as a percentage is 99.97%and the mean left lung dose as a percentage is 86.33%. For both the dailyand average prescient treatments, the minimum tumor dose as a percentage is100%; for the daily prescient treatment, the mean left lung dose as a percentageis 86.18%, whereas for the average prescient treatment, the mean left lung doseas a percentage is 86.02%. Even at slower rates of adaptation, the adaptiverobust treatments perform similarly to the prescient solutions. For instance,using exponential smoothing with α = 0.1 and starting with the robust (R)uncertainty set (implementation (ES(0.1),R)), the minimum tumor dose as apercentage of 72Gy is within 0.5% of the prescient solutions in minimum tu-mor dose and within 1.5% of the prescient solutions in mean left lung dose.Although our present computational results provide a specific example of thisphenomenon, and our analysis in Section 6 provides theoretical justification forthis phenomenon under quite general circumstances, further testing on otherpatient cases is needed to verify the degree to which this insight holds morebroadly in practice.

The third insight is that the final dose distribution from any of the adaptiverobust methods is far less sensitive to the choice of initial uncertainty set com-pared to the static method. For instance, for the first PMF sequence, considerthe treatments that use exponential smoothing with α = 0.9 (implementations(ES(0.9),N), (ES(0.9),R) and (ES(0.9),M)). For the nominal (N), robust (R)and margin (M) uncertainty sets, the minimum tumor doses achieved using ex-ponential smoothing with α = 0.9 are 71.83, 71.98 and 72.02Gy respectively,while the mean left lung doses are 17.57, 17.60 and 17.71Gy respectively. Vi-sually, this insight is evident in the fact that the adaptive method frontiers aremuch smaller than the static frontier on both Figures 1. This insight is signif-

17

icant because it indicates that the success of this method is not contingent onhaving accurate knowledge of what the PMFs will be before the treatment be-gins. From a clinical standpoint, this eliminates an important barrier to clinicalimplementation, as the treatment planner may not have enough detailed knowl-edge of a particular patient’s breathing patterns before the start of treatmentto create a single uncertainty set suitable for the entire treatment course. More-over, the method proposed in Bortfeld et al. (2008) for the design of uncertaintysets requires historical breathing data from prior patients that is similar to thebreathing data of the current patient, which may not always be available.

Given the third insight, it may seem that adding adaptation diminishes thevalue of robust optimization, as the adaptive treatments that begin with a nom-inal uncertainty set have similar performance to the adaptive treatments thatbegin with the other two larger uncertainty sets. Furthermore, the treatmentsfrom exponential smoothing with α = 1 ((ES(1),N), (ES(1),R) and (ES(1),M)),for which the uncertainty set from fraction 2 on is always a singleton, performalmost as well as the prescient solutions with respect to the final dose distri-bution. Although this may be true for the PMF sequences that we study here,it may not be true in general, as the PMF sequences that we have used hereare well-behaved and relatively stable. For instance, sequences that exhibitmore variability may cause poor performance if there is too much adaptationand too little robustness (i.e., α is set close to 1 and the initial uncertaintyset is small); we explore such a sequence in Section C of the Online Supple-ment. Similarly, sequences that exhibit transient behavior and drift outside ofthe initial uncertainty set towards a new value could also necessitate the useof a large uncertainty set early on while the lower and upper bound vectorsare still “homing in” on the PMFs. The value of the methodology that we havedescribed in this paper is that it allows the treatment planner to balance robust-ness, which provides protection during the early fractions of the treatment, andadaptation, which provides protection in the later fractions of the treatment.Thus, this methodology is flexible enough to be applied to settings where thepatient’s breathing will be very stable over the course of the treatment (suchas the PMF sequences that we study here) and settings where the patient’sbreathing could be highly variable early in the treatment or even throughoutthe entire treatment (such as the PMF sequence studied in Section C of the On-line Supplement). Lastly, our computational results correspond to a standardtreatment schedule of 30 fractions. When the number of fractions is reduced,as in hypofractionated treatments, a larger proportion of the total prescriptiondose needs to be delivered in each fraction. In such treatments, the consequencesof selecting an inappropriate uncertainty set in each fraction therefore becomemore severe. Furthermore, the errors in delivered dose made in the early frac-tions can no longer be “averaged out” due to a large number of fractions. Weexpect that in such hypofractionated treatments, the quality of the treamentsresulting from our method will be more strongly tied to the choice of initialuncertainty set. Thus, the performance of the adaptive robust method (usinga non-singleton uncertainty set) will be further differentiated in a positive wayfrom the performance of a nominal (no robustness) adaptive method.

18

6. Asymptotic optimality of the adaptive robust method

In Section 5.3, we observed that the adaptive robust treatments are a) gen-erally better than the static robust treatments in both lung dose and tumorcoverage, b) very close in quality to both prescient solutions, and c) relativelyinsensitive to the choice of initial uncertainty set. In this section, we describeour core theoretical result, which we use to explain these observations. Ourresult is as follows: if the sequence of breathing motion PMFs (pi)∞i=1 convergesto p∗ and the uncertainty set update algorithm belongs to a special class ofupdate algorithms, then as the number of fractions n tends to infinity, the dosedistribution eventually enters the epsilon neighborhood of a set D of “ideal”dose distributions. The set D is the set of dose distributions obtained when anyoptimal solution to the nominal problem with respect to p∗ is delivered whilethe patient is actually breathing according to p∗. The proof of this statement,along with the assumptions and the auxilliary results leading to it, is given inSection D of the Online Supplement.

We define the epsilon neighborhood U(V, ε) of a subset V ⊆ Rn as

U(V, ε) =⋃x∈V

B(x, ε),

where B(x, ε) is the open ball of radius ε about x (B(x, ε) = {x′ ∈ Rn ‖x−x′‖ <ε}). The norm ‖·‖ is the 1-norm on the appropriate finite dimensional real vectorspace, although technically any p-norm can be used.

We say that an uncertainty set update algorithm is a convex-convergentupdate algorithm if it satisfies two conditions:

1. For every i ∈ Z+ (the set of positive integers), `i+1 and ui+1 can bewritten as convex combinations of pi with `i and ui respectively; that is,there exists an αi ∈ [0, 1] such that

`i+1 = (1− αi)`i + αip

i,

ui+1 = (1− αi)ui + αip

i.

2. If pi → p∗ as i→∞, then `i → p∗ and ui → p∗ as i→∞.

It can be easily verified that both the exponential smoothing and running aver-age update algorithms are convex-convergent.

Let w∗(`,u) denote the set of optimal solutions for the robust problem (1)with its uncertainty set P defined by lower and upper bound vectors ` and u.Let w∗(p∗) denote the set of optimal solutions for the nominal problem withrespect to p∗.

Define the set D, the set of dose distributions when p∗ is realized whilew ∈ w∗(p∗) is being delivered, as

D ={d ∈ R|V| d = ∆p∗w for some w ∈ w∗(p∗)

}

19

where the product ∆pw is defined as

∆pw =

[∑x∈X

∑b∈B

∆v,x,bp(x)wb

]v∈V

.

For an n fraction treatment during which the intensity vectors were w1, . . . ,wn

and the realized PMFs were p1, . . . ,pn, the final dose distribution is given by1/n ·

∑ni=1 ∆piwi. Our core result can be stated as Theorem 1.

Theorem 1. (Adaptive robust dose convergence.) Let (pi)∞i=1 be a sequence ofPMFs that converges to p∗. Let (`i)∞i=1 and (ui)∞i=1 be lower and upper boundsequences generated from (pi)∞i=1 by a convex-convergent update algorithm. Foreach i ∈ Z+, let w

i ∈ w∗(`i,ui). Then for every ε > 0, there exists an N ∈ Z+

such that for all n > N ,

1

n

n∑i=1

∆piwi ∈ U(D, ε).

The theorem states that the dose distribution obtained via any convex-convergentupdate algorithm approaches a set of optimal dose distributions D. The intu-ition behind this result is that as i → ∞, wi is increasingly similar to somesolution in w∗(p∗), and pi is increasingly similar to p∗. Therefore, as n tendsto infinity, each term in the tail of the sum

∑ni=1 ∆piwi becomes increasingly

similar to some d ∈ D.This set D of optimal dose distributions is the same set that is approached

by both the daily prescient and average prescient dose distributions.

Theorem 2. Let (pi)∞i=1 be a sequence of PMFs that converges to p∗.

(a) (Daily prescient dose convergence.) For each i ∈ Z+, let wi ∈ w∗(pi,pi).

Then for every ε > 0, there exists an N ∈ Z+ such that for all n > N ,

1

n

n∑i=1

∆piwi ∈ U(D, ε).

(b) (Average prescient dose convergence.) For each n ∈ Z+, let wn ∈ w∗(1/n·∑n

i=1 pi, 1/n ·

∑ni=1 p

i). Then for every ε > 0, there exists an N ∈ Z+

such that for all n > N ,

1

n

n∑i=1

∆piwn ∈ U(D, ε).

When the nominal problem with respect to p∗ has a unique optimal solution,the difference between the adaptive robust and either prescient dose distributiontends to zero.

20

Corollary 1. Let (pi)∞i=1 be a sequence of PMFs that converges to p∗. For eachi ∈ Z+, let w

iAR ∈ w∗(`i,ui) (the adaptive robust intensity vector for fraction i),

where (`i)∞i=1 and (ui)∞i=1 are obtained by a convex-convergent update algorithm,and let wi

DP ∈ w∗(pi,pi) (the daily prescient intensity vector for fraction i).For each n ∈ Z+, let wn

AP ∈ w∗(1/n ·

∑ni=1 p

i, 1/n ·∑n

i=1 pi)(the average

prescient intensity vector for fraction i). If the nominal problem with respect top∗ has a unique optimal solution (i.e., w∗(p∗) is a singleton), then

limn→∞

∥∥∥∥∥ 1

n

n∑i=1

∆piwiAR −

1

n

n∑i=1

∆piwiDP

∥∥∥∥∥= lim

n→∞

∥∥∥∥∥ 1

n

n∑i=1

∆piwiAR −

1

n

n∑i=1

∆piwnAP

∥∥∥∥∥ = 0.

6.1. Three theoretical insights

These results directly relate to the three insights of Section 5.3. With regardto the first insight – that adaptive robust treatments generally dominate staticrobust treatments – suppose that w is a solution of the robust problem withuncertainty set P . The static robust dose distribution after n fractions is givenby

1

n

n∑i=1

∆piw = ∆

n∑i=1

pi

nw.

If (pi)∞i=1 converges to p∗, then this dose distribution converges to ∆p∗w.Whether or not this dose distribution is satisfactory depends on whether ornot p∗ is in P . If p∗ ∈ P , then every tumor voxel will receive sufficient dose inthe limiting dose distribution. However, if p∗ /∈ P , then there is no guaranteethat every tumor voxel will sufficient dose. In contrast to the static method, theadaptive robust method always leads to sufficient tumor dose in the limit. Thereason for this is that every dose distribution d in D has the property that alltumor voxels receive the minimum prescription dose. This property is a conse-quence of the fact that d = ∆p∗w for some intensity vector w correspondingto the singleton uncertainty set {p∗}; by the definition of problem (1), it isstraightforward to see that the dose to tumor voxel v satisfies

dv =∑x∈X

∑b∈B

∆v,x,bp∗(x)wb ≥ θv.

Furthermore, observe that every beamlet intensity vector in w∗(p∗) is anoptimal solution of the nominal problem with respect to p∗. As we discussedearlier, solutions to nominal problems are ones that generally result in the lowestlevel of normal tissue damage. Therefore, not only do the dose distributions in Dmeet the tumor dose requirements, but they are also likely to lead to low levelsof healthy tissue dose. In contrast, a static robust treatment will generally notpossess both of these qualities simultaneously. If the treatment planner selects

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P to be a singleton (i.e. w is the solution of the nominal problem with a specificPMF), then the treatment is likely to result in low healthy tissue dose, but thetreatment planner is taking a major risk by assuming what p∗ will be. If thetreatment planner selects a larger P , then there is a better chance that p∗ willbe contained in P , but at the cost of increased healthy tissue dose.

With regard to the second insight – that the adaptive robust and prescienttreatments become very similar – we know that both the average and dailyprescient dose distributions also approach D. When the nominal problem withrespect to p∗ has a unique optimal solution, the difference between the adaptiverobust dose distribution and either prescient dose distribution must tend to zeroby Corollary 1. In the case that there are multiple optimal solutions to thenominal problem with respect to p∗, it seems reasonable to expect that thedose distributions that are achieved when these solutions are delivered and thepatient breathes according to p∗ should not differ significantly. Therefore, thedifference between either prescient dose distribution and the adaptive robustdose distribution should also become small in the limit.

With regard to the third insight – that the adaptive robust method is rela-tively insensitive to the choice of initial uncertainty set – we can see that in theproof of Theorem 1, the choice of `1 and u1 is not important: the dose distribu-tion always approaches the set D, no matter what `1 and u1 are. Again, if Dis a singleton, then any two dose distributions that are obtained using the sameadaptive method but different `1 and u1 vectors will converge to the single dis-tribution in D. Since the two dose distributions converge to the same limitingdose distribution, the difference between those dose distributions must tend tozero. In practice, we will still see some dependence on the initial uncertaintyset due to the finite number of fractions, but with a large number of fractions,the effect of the initial uncertainty set will become diminished.

We note that although the theoretical results we have developed are helpfulin understanding the performance of our methods, it is unlikely that a patientPMF sequence in real life will converge to a single limiting PMF. A more plau-sible scenario may be that after a transient period, during which the patientis being acclimated to the treatment, the PMF sequence converges to a setof PMFs, and on each day the patient’s PMF varies within this set, withoutconverging to a single limiting PMF. Our results cannot directly answer thisquestion without modifications. However, our core theoretical result (Theo-rem 1) relies on the fact that for every ε, which represents the error between theset of ideal dose distributions D corresponding to a limiting p∗ and the realizeddose distribution 1/n ·

∑ni=1 ∆piwi, we are able to get pi within a suitable

distance of p∗. In the case that we can only say that pi gets arbitrarily close toa set of limiting PMFs, we expect that Theorem 1 can be modified to hold forε ≥ ε0, where ε0 > 0 depends on the size of the set of limiting PMFs, as opposedto any arbitrary ε > 0. We expect that as the size of the limiting set of PMFsdecreases, the error between the realized dose distribution and an ideal set ofdistributions (obtained by applying the daily prescient method to the sequence)will also decrease; Theorem 1 would then present an extreme case where thelimiting set of PMFs is a singleton. The real patient PMF sequences used in

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Section 5 are relatively stable, which is why our present theoretical results agreewith our computational insights. We also expect that when the PMFs convergeonly to a set, the performance may be more sensitive to the choice of the updatealgorithm, and how the convergence to a set is preserved (or not preserved) forthe sequence of lower and upper bound vectors generated by the update algo-rithm. In Section C of the Online Supplement, we provide an example of sucha PMF sequence that does not stabilize. For this sequence, we indeed find thatthe performance gap between the exponential smoothing and prescient solutionsis generally quite large; however, the running average algorithm is still able toachieve good performance.

7. Conclusions

In this study, we developed an optimization method for fractionated IMRTtreatment planning that combats breathing motion uncertainty by generalizingthe methodologies of adaptive radiation therapy and robust optimization. Westudied the effectiveness of our method on two PMF sequences obtained fromclinical lung cancer patients and showed that our adaptive robust optimizationmethod improves upon the static method in both tumor coverage and healthytissue sparing simultaneously. We also studied the effectiveness of our methodfrom a theoretical standpoint and proved that the adaptive robust dose distri-bution approaches a limiting set of optimal dose distributions. Even thoughthis adaptive approach is myopic, it performs well when the PMF sequenceconverges, and for some uncertainty set update algorithms, performs well evenwhen the PMF sequence does not converge. Also, as we have shown empiricallyand theoretically, our method does not require perfectly accurate informationto be available before the start of treatment. Overall, the clinical value of thismethod is that it allows for the tumor dose to be safely escalated without lead-ing to additional healthy tissue toxicity, which may ultimately improve the rateof patient survival.

Future work should consider what results can be obtained when the conver-gence assumption is relaxed and the PMFs are instead drawn from a distributionon the probability simplex. The adaptive robust method we have studied canalso be modified by changing the right hand side of problem (1) after eachfraction, in order for the treatment to account for the dose delivered in priorfractions and to compensate for daily underdosages and overdosages in priorfractions. With such a method, we hypothesize that the daily dose delivered tothe tumor will become increasingly inhomogeneous as the treatment progresses,with some parts of the tumor receiving significantly less than the daily pre-scribed amount. This seems to be the case in some preliminary computationaltests and is the subject of ongoing work.

Finally, we note that although we have specifically studied fractionatedIMRT planning under breathing motion uncertainty, the methodology and anal-ysis we have presented here is of value to other practical problems outside of thedomain of healthcare. Specifically, this method can be applied to problems in-volving sequential decision making under uncertainty, when new measurements

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of the uncertain effect are obtained over the planning horizon. This approachmay be particularly attractive when it is computationally inexpensive to solvethe robust optimization problem with the updated uncertainty set at each stage.Overall, the empirical performance and theoretical properties we have shown inthis paper suggest that this type of adaptive robust optimization method couldbe a viable approach for other multi-stage decision making problems under un-certainty.

8. Acknowledgements

The authors thank the referees for their careful reading of the paper andfor their thoughtful comments which have helped to improve the paper. Theauthors thank Thomas Bortfeld of Massachusetts General Hospital for providingthe patient data used in Section 5. The work of both authors was partiallysupported by the Natural Sciences and Engineering Research Council of Canada(NSERC) and the Canadian Institutes of Health Research (CIHR). The workof the second author was partially supported by an NSERC CGS-M award.

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